Colonies are cities that will eventually grow into full cities when they reach 700 pop (each colonist is worth 100 pop.)
When this size is reached, you can build improvements, troops like in any other city in Europe. Depending on the province's suitability for colonization, the population will grow or decrease (African jungle are not very attractive for EUropean, whereas North AMerican plains are).
On the economics level, these provinces provide you with direct income and increase the value of the COT they belong to.
Trade posts do not grow, they are just here to exploit the resource of the province. They only increase the value of the COT they belong to, therefore trade posts are useful only if you send merchants to the COT. Moreover, trade post never allow for improvements' buildings nor is there a port. Trade posts cannot be fortified and an enemy army can burn it to the ground just like this.
On the other hand, trade posts are easier and cheaper to install (5 gold for a TP versus 50-100 gold for a colony).